Finite Group Databasebeta

← C3×#14027 #10632634

Subgroups — C3×#14027 #10632634

Order 768.

2574 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#124C211yesmaximal, characteristic
2C22×C3×C4×D4C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C2×C3×#391C211yesmaximal, characteristic, complemented
5C2×C3×#390C211yesmaximal, characteristic
6C3×#1440C214yesmaximal, complemented
7C3×C4⋊#95C211yesmaximal, characteristic, complemented
8C3×C22⋊#95C214yesmaximal, complemented
9C3×C4×#95C211yesmaximal, characteristic, complemented
10#14027C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C24×C3×C4C2212yes—
14C2×C3×#124—24no—
15C2×C3×#124—24no—
16C2×C3×#124C418yescomplemented
17C2×C3×C4×D4—24no—
18C2×C3×C4×D4—24no—
19C2×C3×C4×D4—24no—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2214yes—
24C22×C3×C22⋊C4C2214yes—
25C22×C3×C42C2211yescharacteristic
26C22×C3×C42C2211yescharacteristic
27C22×C3×C42C2211yescharacteristic
28C3×#391—24no—
29C3×#391—24no—
30C3×#390—24no—
31C2×C3×#95C2212yes—
32C2×C3×#95C2214yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2214yes—
37C22×#124C2×C311yescharacteristic
38C22×C4×D4C2×C311yescharacteristic
39C2×#391C2×C311yescharacteristic, complemented
40C2×#391C2×C311yescharacteristic, complemented
41C2×#390C2×C311yescharacteristic
42#1440C2×C314yescomplemented
43C4⋊#95C2×C311yescharacteristic, complemented
44C22⋊#95C2×C314yescomplemented
45C4×#95C2×C311yescharacteristic, complemented
46C25×C3C2312yes—
47C22×C3×Q8—22no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8C2×C414yescomplemented
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4C2×C414yes—
59C22×C3×D4D414yes—
60C22×C3×D4D414yes—
61C23×C3×C4—24no—
62C23×C3×C4—24no—
63C23×C3×C4—24no—
64C23×C3×C4—28no—
65C23×C3×C4—44no—
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C3×#124—232no—
75C3×C4×D4—232no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—28no—
84C2×C3×C22⋊C4—28no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—28no—
89C2×C3×C22⋊C4—44no—
90C2×C3×C22⋊C4C2×C4116yescomplemented
91C2×C3×C42—22no—
92C2×C3×C42—22no—
93C2×C3×C42—22no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42—22no—
97C2×C3×C42—22no—
98C2×C3×C42C2×C414yescomplemented
99C2×C3×C42D414yes—
100C3×#95—28no—
101C3×#95—24no—
102C3×#95—24no—
103C3×#95—24no—
104C3×#95—24no—
105C3×#95—44no—
106C23×Q8C22×C311yescharacteristic
107C23×D4C22×C311yescharacteristic
108C24×C4C22×C312yes—
109C2×#124—24no—
110C2×#124—24no—
111C2×#124C3×C418yescomplemented
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C22×C4⋊C4C22×C311yescharacteristic
116C22×C4⋊C4C22×C311yescharacteristic
117C22×C22⋊C4C22×C312yes—
118C22×C22⋊C4C22×C314yes—
119C22×C22⋊C4C22×C314yes—
120C22×C42C22×C311yescharacteristic
121C22×C42C22×C311yescharacteristic
122C22×C42C22×C311yescharacteristic
123#391—24no—
124#391—24no—
125#390—24no—
126C2×#95C22×C312yes—
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C2×#95C22×C314yes—
130C2×#95C22×C312yes—
131C2×#95C22×C314yes—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—24no—
135C24×C3—24no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—24no—
139C24×C3—24no—
140C24×C3C22×C414yes—
141C24×C3C2×D414yes—
142C24×C3C2×D414yes—
143C24×C3C2411yescharacteristic
144C2×C3×Q8—28no—
145C2×C3×Q8—28no—
146C2×C3×Q8—28no—
147C2×C3×Q8—44no—
148C2×C3×Q8—44no—
149C2×C3×Q8—44no—
150C2×C3×Q8—44no—
151C2×C3×D4—28no—
152C2×C3×D4—28no—
153C2×C3×D4—28no—
154C2×C3×D4—28no—
155C2×C3×D4—28no—
156C2×C3×D4—28no—
157C2×C3×D4C22⋊C4116yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—216no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—24no—
193C22×C3×C4—416no—
194C22×C3×C4—416no—
195C22×C3×C4—48no—
196C22×C3×C4—48no—
197C22×C3×C4C22×C414yes—
198C22×C3×C4C22×C414yes—
199C22×C3×C4C22×C412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D414yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4#4114yes—
208C22×C3×C4#4114yes—
209C22×C3×C4#4112yes—
210C22×C3×C4#4112yes—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—44no—
213C3×C22⋊C4—216no—
214C3×C22⋊C4—232no—
215C3×C22⋊C4—416no—
216C3×C42—28no—
217C3×C42—28no—
218C3×C42—44no—
219C25C23×C312yes—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8C2×C3×C414yescomplemented
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C2×C3×C414yes—
232C22×D4C3×D414yes—
233C22×D4C3×D414yes—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—28no—
238C23×C4—44no—
239C23×C4C23×C312yes—
240C23×C4C23×C312yes—
241C23×C4C23×C312yes—
242C23×C4C23×C311yescharacteristic
243C23×C4C23×C312yes—
244C23×C4C23×C312yes—
245C23×C4C23×C311yescharacteristic
246C23×C4C23×C311yescharacteristic
247#124—232no—
248C4×D4—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C3×D414yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C2×C3×C4116yescomplemented
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42C2×C3×C414yescomplemented
272C2×C42C3×D414yes—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—44no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—28no—
282C23×C3—28no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—28no—
288C23×C3—28no—
289C23×C3—28no—
290C23×C3—28no—
291C23×C3—42no—
292C23×C3—44no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3C2×C22⋊C418yes—
302C23×C3C23×C411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C22×D411yescharacteristic
306C23×C3C22×D411yescharacteristic
307C23×C3C2×#4111yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3H(5,𝔽2)12yes—
312C23×C32−1+412yes—
313C3×Q8—416no—
314C3×D4—232no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—28no—
332C2×C3×C4—24no—
333C2×C3×C4—432no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4C2×C22⋊C414yes—
357C2×C3×C4C4×D414yes—
358C2×C3×C4C4×D414yes—
359C2×C3×C4#12014yes—
360C2×C3×C4#12114yes—
361C2×C3×C4#12114yes—
362C2×C3×C4#12314yes—
363C2×C3×C4#12414yescomplemented
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24C22×C3×C414yes—
373C24C2×C3×D414yes—
374C24C2×C3×D414yes—
375C24C24×C311yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4C3×C22⋊C4116yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4C22×C3×C414yes—
430C22×C4C22×C3×C414yes—
431C22×C4C22×C3×C412yes—
432C22×C4C2×C3×D412yes—
433C22×C4C2×C3×D414yes—
434C22×C4C2×C3×D412yes—
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D412yes—
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D412yes—
439C22×C4C3×#4114yes—
440C22×C4C3×#4114yes—
441C22×C4C3×#4112yes—
442C22×C4C3×#4112yes—
443C4⋊C4—28no—
444C4⋊C4—44no—
445C22⋊C4—216no—
446C22⋊C4—232no—
447C22⋊C4—416no—
448C42—28no—
449C42—28no—
450C42—44no—
451C22×C3—216no—
452C22×C3—28no—
453C22×C3—44no—
454C22×C3—48no—
455C22×C3—48no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—48no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3C22×C22⋊C411yescharacteristic
463C22×C3C2×C4×D411yescharacteristic
464C22×C3C2×C4×D411yescharacteristic
465C22×C3C4×#4112yes—
466C22×C3#52012yes—
467C22×C3C2×#12011yescharacteristic
468C22×C3C2×#12111yescharacteristic
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×#12311yescharacteristic
471C22×C3C2×#12411yescharacteristic
472C22×C3#52912yes—
473C22×C3#53211yescharacteristic
474C22×C3#53511yescharacteristic
475C22×C3#53611yescharacteristic
476C22×C3#54212yes—
477C22×C3#54612yes—
478C22×C3#54612yes—
479C22×C3#54612yes—
480C22×C3#54712yes—
481C22×C3#54712yes—
482C22×C3#54812yes—
483C22×C3#55612yes—
484C22×C3#55912yes—
485C3×C4—24no—
486C3×C4—44no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4—44no—
491C3×C4—82no—
492C3×C4#38618yes—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23C2×C3×C22⋊C418yes—
516C23C23×C3×C411yescharacteristic
517C23C22×C3×D411yescharacteristic
518C23C22×C3×D411yescharacteristic
519C23C22×C3×D411yescharacteristic
520C23C22×C3×D411yescharacteristic
521C23C2×C3×#4111yescharacteristic
522C23C2×C3×#4112yes—
523C23C2×C3×#4111yescharacteristic
524C23C2×C3×#4112yes—
525C23C3×H(5,𝔽2)12yes—
526C23C3×2−1+412yes—
527Q8—416no—
528D4—232no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—432no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4C2×C3×C22⋊C414yes—
571C2×C4C3×C4×D414yes—
572C2×C4C3×C4×D414yes—
573C2×C4C3×#12014yes—
574C2×C4C3×#12114yes—
575C2×C4C3×#12114yes—
576C2×C4C3×#12314yes—
577C2×C4C3×#12414yescomplemented
578C2×C3—48no—
579C2×C3C2×#38611yescharacteristic
580C2×C3C4×#12411yescharacteristic
581C2×C3#232611yescharacteristic
582C2×C3#233912yes—
583C2×C3#236712yes—
584C2×C3#242712yes—
585C2×C3#243211yescharacteristic
586C2×C3#243511yescharacteristic
587C2×C3#244712yes—
588C2×C3#254411yescharacteristic
589C2×C3#254711yescharacteristic
590C22—28no—
591C22—216no—
592C22—48no—
593C22—48no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22C22×C3×C22⋊C411yescharacteristic
602C22C2×C3×C4×D411yescharacteristic
603C22C2×C3×C4×D411yescharacteristic
604C22C3×C4×#4112yes—
605C22C3×#52012yes—
606C22C2×C3×#12011yescharacteristic
607C22C2×C3×#12111yescharacteristic
608C22C2×C3×#12111yescharacteristic
609C22C2×C3×#12311yescharacteristic
610C22C2×C3×#12411yescharacteristic
611C22C3×#52912yes—
612C22C3×#53211yescharacteristic
613C22C3×#53511yescharacteristic
614C22C3×#53611yescharacteristic
615C22C3×#54212yes—
616C22C3×#54612yes—
617C22C3×#54612yes—
618C22C3×#54612yes—
619C22C3×#54712yes—
620C22C3×#54712yes—
621C22C3×#54812yes—
622C22C3×#55612yes—
623C22C3×#55912yes—
624C4—24no—
625C4—44no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4C3×#38618yes—
632C3#1402711yescharacteristic, direct product factor, Sylow
633C2—48no—
634C2C2×C3×#38611yescharacteristic
635C2C3×C4×#12411yescharacteristic
636C2C3×#232611yescharacteristic
637C2C3×#233912yes—
638C2C3×#236712yes—
639C2C3×#242712yes—
640C2C3×#243211yescharacteristic
641C2C3×#243511yescharacteristic
642C2C3×#244712yes—
643C2C3×#254411yescharacteristic
644C2C3×#254711yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C3×#391C2semidirect
4. C2×C3×#391C2semidirect
6. C3×#1440C2semidirect
7. C3×C4⋊#95C2semidirect
8. C3×C22⋊#95C2semidirect
9. C3×C4×#95C2semidirect
10. #14027C3direct (N × H)
16. C2×C3×#124C4semidirect
39. C2×#391C2×C3semidirect
40. C2×#391C2×C3semidirect
42. #1440C2×C3semidirect
43. C4⋊#95C2×C3semidirect
44. C22⋊#95C2×C3semidirect
45. C4×#95C2×C3semidirect
53. C22×C3×Q8C2×C4semidirect
90. C2×C3×C22⋊C4C2×C4semidirect
98. C2×C3×C42C2×C4semidirect
111. C2×#124C3×C4semidirect
226. C22×Q8C2×C3×C4semidirect
263. C2×C22⋊C4C2×C3×C4semidirect
271. C2×C42C2×C3×C4semidirect
363. C2×C3×C4#124semidirect
577. C2×C4C3×#124semidirect
632. C3#14027direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×#1242
C22×C3×C4×D42
C2×C3×#3912
C2×C3×#3902
C3×#14402
C3×C4⋊#952
C3×C22⋊#952
C3×C4×#952
#140273